Real Analysis : An Introduction to the Theory of Real Functions and Integration

Designed for use in a two-semester course on abstract analysis, REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis. Self-contained, with coverage of topology, measure theory, and integration, it offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering. Structured logically and flexibly through the author's many years of teaching experience, the material is presented in three main sections: Part 1, chapters 1through 3, covers the preliminaries of set theory and the fundamentals of metric spaces and topology. This section can also serves as a text for first courses in topology. Part II, chapter 4 through 7, details the basics of measure and integration and stands independently for use in a separate measure theory course. Part III addresses more advanced topics, including elaborated and abstract versions of measure and integration along with their applications to functional analysis, probability theory, and conventional analysis on the real line. Analysis lies at the core of all mathematical disciplines, and as such, students need and deserve a careful, rigorous presentation of the material. REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration offers the perfect vehicle for building the foundation students need for more advanced studies. Preface......Page 7 Contents......Page 11 Part I: An Introduction to General Topology......Page 15 1. Sets and Basic Notation......Page 17 2. Functions......Page 25 3. Set Operations under Maps......Page 31 4. Relations and Well-Ordering Principle......Page 36 5. Cartesian Product......Page 45 6. Cardinality......Page 54 7. Basic Algebraic Structures......Page 60 1. Defmitions and Notations......Page 73 2. The Structure of Metric Spaces......Page 79 3. Convergence in Metric Spaces......Page 88 4. Continuous Mappings in Metric Spaces......Page 92 5. Complete Metric Spaces......Page 101 6. Compactness......Page 106 7. Linear and Normed Linear Spaces......Page 114 1. Topological Spaces......Page 121 2. Bases and Subbases for Topological Spaces......Page 129 3. Convergence of Sequences in Topological Spaces and Countability......Page 136 4. Continuity in Topological Spaces......Page 142 5. Product Topology......Page 149 6. Notes on Subspaces and Compactness......Page 157 7. Function Spaces and Ascoli's Theorem......Page 165 8. Stone-Weierstrass Approximation Theorem......Page 174 9. Filter and Net Convergence......Page 181 10. Separation......Page 196 11. Functions on Locally Compact Spaces......Page 209 Part II: Basics of Measure and Integration......Page 215 4 Measurable Spaces and Measurable Functions......Page 217 1. Systems of Sets......Page 218 2. System's Generators......Page 224 3. Measurable Functions......Page 230 5 Measures......Page 235 1. Set Functions......Page 236 2. Extension of Set Functions to a Measure......Page 249 3. Lebesgue and Lebesgue-Stieltjes Measures......Page 272 4. Image Measures......Page 291 5. Extended Real-Valued Measurable Functions......Page 296 6. Simple Functions......Page 302 6 Elements of Integration......Page 309 1. INTEGRATION ON e^(-1)(Omega,E)......Page 310 2. Main Convergence Theorems......Page 326 3. Lebesgue and Riemann Integrals on R......Page 341 4. Integration with Respect to Image Measures......Page 355 5. Measures Generated by Integrals. Absolute Continuity. Orthogonality......Page 360 6. Product Measures of Finitely Many Measurable Spaces and Fubini's Theorem......Page 370 7. Applications ofFubini's Theorem......Page 392 1. Differentiation......Page 401 2. Change of Variables......Page 416 Part III. Further Topics in Integration......Page 433 8 Analysis in Abstract Spaces......Page 435 1. Signed and Complex Measures......Page 436 2. Absolute Continuity......Page 451 3. Singularity......Page 466 4. L^P Spaces......Page 474 5. Modes of Convergence......Page 488 6. Uniform Integrability......Page 500 7. Radon Measures on Locally Compact Hausdorff Spaces......Page 507 8. Measure Derivatives......Page 524 1. Monotone Functions......Page 531 2. Functions of Bounded Variation......Page 542 3. Absolute Continuous Functions......Page 549 4. Singular Functions......Page 557 Bibliography......Page 565 INDEX......Page 567 Real Analysis: An Introduction To The Theory Of Real Functions And Integration Illuminates The Principal Topics That Constitute Real Analysis. Self-contained, With Coverage Of Topology, Measure Theory, And Integration, It Offers A Thorough Elaboration Of The Major Theorems, Notions, And Constructions Needed Not Only By Those Interested In The Mathematical Sciences But Also By Those Pursuing Careers In Statistics And Probability, Operations Research, Physics, And Engineering. Real Analysis: An Introduction To The Theory Of Real Functions And Integration Offers The Perfect Vehicle For Building The Foundation Needed For More Advanced Studies.--book Jacket. Set-theoretic And Algebraic Preliminaries -- Analysis Of Metric Spaces -- Elements Of Point Set Topology -- Measurable Spaces And Measurable Functions -- Measures -- Elements Of Integration -- Calculus In Euclidean Spaces -- Analysis In Abstract Spaces -- Calculus On The Real Line. Jewgeni H. Dshalalow. Includes Bibliographical References (p. [551]) And Index. Self-contained, with coverage of topology, measure theory, and integration, this book offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering. With coverage of topology, measure theory and integration, this text offers a thorough elaboration of major theorems, notions and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics and engineering Set theory is not just one of the main tools in mathematics, it is the very root of mathematics, from which all mathematical disciplines stem.